Generalizations of theorems of Nishino and Hartogs by the \(L^2\) method
In [J. Math. Kyoto Univ. 9, 221--274 (1969; Zbl 0192.43703)], \textit{T. Nishino} proved a rigidity result for Stein surfaces: Let \(X\) be a Stein surface with a submersion \(\pi: X\longrightarrow \mathbb{D}\) onto the unit disk in \(\mathbb{C}\) such that every fiber of \(\pi\) is holomorphically equivalent to \(\mathbb{C}\). Then \(\pi\) is holomorphically equivalent to the projection \(\mathbb{C}\times\mathbb{D}\longrightarrow\mathbb{D}\). Different proofs of this result have been obtained by \textit{H. Yamaguchi} [J. Math. Kyoto Univ. 16, 497--530 (1976; Zbl 0343.32012)] and \textit{E. M. Chirka} [Russ. Math. Surv. 67, No. 6, 1091--1165 (2012; Zbl 1266.30027); translation from Usp. Mat. Nauk 67, No. 6, 125--202 (2012)]. In [Kyoto J. Math. 60, No. 3, 1047--1050 (2020; Zbl 1457.32030)], \textit{T. Ohsawa} published yet another proof of Nishino's theorem based on an \(L^2\) extension theorem. In the present paper, the author uses \(L^2\) vanishing theorems instead of \(L^2\) extension theorems to obtain generalizations of Nishino's rigidity theorem as follows: Theorem 0.1. Let \(M\) be a complex manifold of dimension \(m + 1\) which admits a complete Kähler metric and a holomorphic map \(\pi\) onto \(\mathbb{D}^m\) without critical points such that \(\pi^{-1}(t) \simeq\mathbb{C}\) for all \(t\in\mathbb{D}^m\). Then \(M\) is biholomorphically equivalent to \(\mathbb{D}^m\times\mathbb{C}\). Theorem 0.2. Let \(M\) be a complex manifold of dimension \(m + n\) with a holomorphic submersion onto \(\mathbb{D}^m\) whose fibers are once-punctured \(\mathbb{CP}^n\). Then \(M \simeq (\mathbb{CP}^n)^\ast\times \mathbb{D}^m\) if and only if \(M\) is \(n\)-convex. As a corollary, one has the following result, that was independently obtained by \textit{T. P. Pawlaschyk} [On some classes of \(q\)-plurisubharmonic functions and \(q\)-pseudoconcave sets. Wuppertal: Bergische Universität Wuppertal (Diss.) (2015)]: Corollary 0.1. Let \(f : \mathbb{D}^m\longrightarrow \mathbb{C}^n\) be a continuous function. Then \(f\) is holomorphic if and only if \((\mathbb{D}^m\times\mathbb{C}^n)\setminus \lbrace (t,f(t)); t\in \mathbb{D}^m\) is \(n\)-convex. Also some global version of Nishino's theorem is proved (Theorem 4.2). Some related open questions are discussed at the end of the paper.
scientific article; zbMATH DE number 7317617
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